A bit simplified:
The Racing Rules of Sailing are a private contract between racing boats and an organizing authority in which the competitors agree to replace the standard IRPCAS/COLREGS or Inland Rules with the RRS. The RRS are designed to allow for boats being very close together yet still maneuver in a safe and predictable manner. The COLREGS by contrast strongly discourage close maneuvering and expect boats to make early and substantial action to eliminate risk of collision.
(The RRS apply ONLY among the competitors. The regular COLREGS apply to encounters with non-racers. In special cases of high-level regattas such as the AC, the organizers may be able to have a race course closed to non-competitors, but that depends on proper arrangements being made with local authorities, and is not typical for most racing. In areas with heavy traffic or special hazards, local authorities may impose special requirements on the race organizers and boats.)
Under the RRS, the leeward boat is in some circumstances limited to a "proper course" (if overlap is acquired after the start by the action of the leeward boat from astern and within two boat lengths), but this is not generally a huge limitation, and this is not just one course, but any course that would be the fast way around the course in the absence of any other boats. At any given time, a boat may have multiple "proper courses" from which to choose.
Under the RRS, the leeward boat can luff up the windward boat, but upon leeward acquiring right of way, she must initially give windward an opportunity to keep clear. This opportunity is limited in time and windward must respond to leeward's luff if it is possible for her to do so. Exceptions might be if there are obstructions/hazards immediately to windward of the windward boat.
Contact between boats is discouraged under the RRS, but does not necessarily result in a penalty.
There are some fine points of difference between the general fleet racing rules in the RRS, and the special rules for match racing. The most visible difference is the procedure of a pre-start entry into a "starting box". There are also some other bits that bring the match racing RRS even further from the COLREGS and allow for more intense and aggressive maneuvering than would be acceptable in a large fleet.
Also, match racing tends to be the sphere of action of highly skilled sailors, with the America's Cup being an extreme case. AC sailors are expected to be capable of a high level of seamanship and to be able to safely execute close-quarters tactical "moves" that are a standard part of the canon of match racing. This is still true in spite of the fact that the crews in this year's Cup were also very busy mastering extreme high-performance boats.
A fun article,
Fleet and Match Racing Rules Compared | Sailing World